Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Struktura elektronowa i dynamika sieciczystego i zdefektowanego CoO

Urszula D. Wdowik, Krzysztof Parlinski

Wydzial Matematyczno-Fizyczno-Techniczny,Uniwersytet Pedagogiczny, Krakow

Seminarium WFiIS 3/04/2009

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Outline

1 Introduction

structure, electronic and magnetic properties of CoO2 Methodology

3 Results for defect-free CoO

4 Phonons (direct method)

5 Cobalt-deficient CoO

electronic structurelattice dynamics

6 Fe-doped CoO

lattice dynamicscharge and spin states of Fe in CoO and vacancy-defectedCoOelectronic structure

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Structure, electronic and magnetic properties of CoO

CoO structure

High temperature structure

Paramagnetic (Fm3m)

aexp = 4.26 Å

Low temperature structure

distorted NaCl structure(R3m) below TN = 293 K

rhombohedral distortionalong <111>, 0.2 − 0.3

2nd kind ofantiferromagnetic ordering,AFII structure

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Structure, electronic and magnetic properties of CoO

Electronic and magnetic properties of CoO

Experiment

Insulator, Eg = 2.5 − 2.8 eV (experiment)

Mtotal = 3.8 − 3.98µB

Theory (DFT)

Charge-transfer insulator

Strong on-site Coulomb interactions in 3d shell cannot beneglected (strongly correlated electron system)

DFT+U formalism required

Interactions between correlated states - Hubbard potentialU and the exchange interaction J

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Calculations with +U formalism

Pseudopotential plane-wave method - VASP code

spin-polarized DFT for AFII structure (64-atom supercell)

Projector-augmented wave pseudopotentials (PAWs)

Exchange-interaction: GGA+U

Full relaxations of AFII structure

U = ?, J = ?

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Energy gap, spin magnetic moment, lattice constant vs U

4.20

4.22

4.24

4.26

4.28

4.30

a (1

0-10 m

)a(exp.) = 4.26x10-10 m

2.30

2.50

2.70

2.90

M (

µ B)

M(exp.) = 3.98 µB

0.000.501.001.502.002.503.00

1 2 3 4 5 6 7 8 9

Eg

(eV

)

U (eV)

Eg(exp.) = 2.5-2.8 eV

Resultsa - weak dependence upon U

MS changes by ca. 17%

Underestimation of Eg for U < 5 eV

Eg vanishes for U = 1 eV (GGA limit)

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Defect-free CoO. Results for U = 7.1 eV, J = 1 eV

-40

-20

0

20

40

DO

S (

stat

es/e

V)

Co t2gCo eg

-20 0

20

-8 -6 -4 -2 0 2 4 6

Energy (eV)

O 2p

U = 7.1 eV, J = 1 eV

a = 4.27ÅRhombohedral distortionalong <111> (0.3o)

space group D53d (R3m)

Eg = 2.77 eV

Ms = 2.74µB

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Method to calculate phonons

Harmonic approximationDirect method - PHONON software by K. Parlinski

displace crystallographically nonequivalent atoms from theirequilibrium positionscalculate Hellmann-Feynman forces F(n, µ) = −δE

δR(n,µ)

Fi(n, µ) = −∑

m,ν,j Φij(n, µ, m, ν)Uj(m, ν)solve dynamical matrix D(k)ω

2(k, j)e(k, j) = D(k)e(k, j)⇓

ω(k, j) - phonon frequenciese(k, j) - polarization vectors of phonons

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Phonon dispersion relations for U = 0 eV and U = 3 eV

Ueff = 0 eV

-2

0

2

4

6

8

10

12

14

Fre

quen

cy (

TH

z)

Γ X K Γ L W X

[ξ00] [ξξ0] [ξξξ ]

0 0.1 0.2

-2

0

2

4

6

8

10

12

14

DOS (1/THz)

Ueff = 2 eV

-4

-2

0

2

4

6

8

10

12

14

16

Fre

quen

cy (

TH

z)

Γ X K Γ L W X

[ξ00] [ξξ0] [ξξξ ]

0 0.2 0.4-4

-2

0

2

4

6

8

10

12

14

16

DOS (1/THz)

imaginary frequencies of acoustic modes ⇒ instability of such CoOstructure

significantly underestimated HF forces ⇒ artificial mode softening

small Ueff ⇒ too low repulsion in 3d shell

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Dispersion relations and phonon DOS for U = 7 eV

U = 7.1 eV, J = 1 eV

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

Γ X K Γ L W X

[ξ00] [ξξ0] [ξξξ ]

0 0.15 0.30

2

4

6

8

10

12

14

16

18

DOS (1/THz)

Data for LO-TO splittingD(k) =D0(k) + DN(ǫ∞, Z∗

, . . . )

high-frequency dielectricconstantǫ∞ = 5.3[1]

Born effective charges

|Z ∗| = 2.06

Experimental data from neutron scattering (T = 110 K)

[1] J. Sakurai, W.J.L. Buyers, R.A. Cowley, and G. Dolling, Phys. Rev. 167, 510 (1968)

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Comparison with experiment

Experiment vs theory

Γ-point frequencies Neutron Infrared Theory(THz) scattering[1] spectroscopy[2]

ωTO 10.50 10.40-10.50 10.25ωLO 15.75 16.30-16.40 15.73

Experimental data

[1] J. Sakurai, W.J.L. Buyers, R.A. Cowley, and G. Dolling, Phys. Rev. 167, 510 (1968)[2] J.P. Gielisse, J.N. Plendl, L.C. Mansur, R. Marshall, S.S. Mitra, R. Mikolajewicz, andA. Smakula, J. Appl. Phys. 36 2426 (1965)

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Heat capacity and Debye-Waller factors

Heat capacity

0

10

20

30

40

50

60

70

0 100 200 300 400 500 600 700 800 900 1000

Hea

t cap

acity

(J/

mol

K)

Temperature (K)

Lattice contribution

magnetic contribution below TN

Dulong-Petit law @ HT

Debye-Waller factors

0.000

0.003

0.006

0.009

0.012

0.015

0.018

0 100 200 300 400 500 600 700 800 900 1000

Mea

n-sq

uare

d di

spla

cem

ents

(10

-20 m

2 )

Temperature (K)

Cobalt (γ-ray)Cobalt (x-ray)

Oxygen (γ-ray)Oxygen (x-ray)

57Fe in CoO

calculated ΘD = 500 K

Mossbauer ⇒ ΘD = 440 K

[3] E.G. King nad A.U. Christensen, U.S. Bur.Mines. Tech. Paper 80, 1800 (1956); E.G. King, ibid. 80, 2399 (1956)[4] W. Jauch and M. Reehuis, PRB 65, 125111 (2002)

[5] S. Sasaki F. Fujino, and K. Takeuchi, Proc. Jpn. Acad. B 55, 43 (1979)[6] K.Ruebenbauer and U.D. Wdowik, J. Phys. Chem. Solids 65, 1785 (2004)

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Electronic structure of CoO defected by cationic vacancies

Is CoO a stoichiometric material?

Native defects in CoO - cationic vacanciesCo deficiency in ’almost stoichiometric’ samples: 0.1-3%

uncharged, singly and doubly charged vacancies

nonstoichiometry depends on temperature and oxygenpartial pressure

How to simulate point defects (vacancies, impurities)

supercell approach

remove atoms ⇒ vacancies

replace host atoms by different kind of atoms ⇒ impurities

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Electronic structure of CoO defected by cationic vacancies

Modeling of non-stoichiometric CoO

Co0.97O

Co @ (12 ,

12 ,

12) removed

from 64-atom SC⇓CoO with 3% vacancies

Co0.94O

Co @ (12 ,

12 ,

12) and (1

2 ,12 , 0)

removed from 64-atom SC⇓CoO with 6% vacancies

Atomic relaxation in defected systems

oxygens surrounding vacancies displace outward vacancysite by 0.12 Å

initial Co-O distance (2.15 Å) changes by ca. 0.01 Å

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Electronic structure of CoO defected by cationic vacancies

Trivalent cations in CoO

Pure CoOMS = 2.74µB

valence charges qCo = +1.33e, qO = −1.33e

Co2+-O 2.15 Å

Co0.97O

MS = 3.15µB

qCo = +1.65e

Co3+ − O 2.05 Å

2 Co3+ in SC

Co0.94O

MS = 3.16µB

qCo = +1.68e

Co3+ − O 2.05 Å

4 Co3+ in SC

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Electronic structure of CoO defected by cationic vacancies

Location of trivalent cations

Co0.97O

-2.0

-1.0

0.0

1.0

2.0

2 3 4 5 6 7 8

Val

ence

cha

rge

(e)

(a)

← Co3+ Co3+ →

-1.0

0.0

1.0

2.0

3.0

4.0

2 3 4 5 6 7 8

Mag

netic

mom

ent (

µ B)

R (Angstrom)

(b)

← Co3+ Co3+ →

Co3+ distance from vacancy

2.96 Å 7.41 Å

Co0.94O

-2.0

-1.0

0.0

1.0

2.0

2 3 4 5 6 7 8

Val

ence

cha

rge

(e)

(a)

Co3+ → ← Co3+

-1.0

0.0

1.0

2.0

3.0

4.0

2 3 4 5 6 7 8

Mag

netic

mom

ent (

µ B)

R (Angstrom)

(b)

Co3+ → ← Co3+

Co3+ distance from vacancy

4.29 Å 6.04 Å

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Electronic structure of CoO defected by cationic vacancies

Electronic structure of vacancy-defected CoO

Co0.97O

-40

-20

0

20

40

DO

S (

stat

es/e

V)

Co t2gCo eg

-20

0

20

-8 -6 -4 -2 0 2 4 6

Energy (eV)

O 2p

Co0.94O

-40

-20

0

20

40

DO

S (

stat

es/e

V)

Co t2gCo eg

-20

0

20

-10 -8 -6 -4 -2 0 2 4 6

Energy (eV)

O 2p

charge transfer from Co to O (via s,p states)

Co2+ ⇒ Co3+ + e−

2 holes associated with vacancy are compensated by 2 Co3+

acceptor states in Co0.97O

1 eV above VBM

acceptor states in Co0.94O

1.5 eV above VBM

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Lattice dynamics of cobalt-deficient CoO

Description of phonons within SC approach

CoO

primitive UCN = 2 ⇒ 6 branches

CoO as 64 atom SC

NS = 64 ⇒ 192 branches ?

Co0.97O

SC - already primitive UCNS = 63 ⇒ 189 branches ?

dimensions of D(k) increases to 3NS

number of phonon dispersion curves increases to 3NS

size of BZ conjugated with SC shrinks

selection of a different kind of unit cell⇓number of phonon dispersion curves is blown up.

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Lattice dynamics of cobalt-deficient CoO

Phonon form factor and filter

Phonon form factor

F (p)(k, j) = 1k2

∣

∣

∣

∣

∑

µ

k·e(k,j ;µ)√Mµ

∣

∣

∣

∣

2

∫

Ω dΩ F (p)(k, j) = 13 F (s)(k, j)

Filter

F (s)(k, j) =

∣

∣

∣

∣

∑

µ

e(k,j ;µ)√Mµ

∣

∣

∣

∣

2

Fake phonon modes

F (p)(k, j) = 0

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Lattice dynamics of cobalt-deficient CoO

Application of filter to stoichiometric CoO

Intensities of modes for CoO

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

X K Γ L W X Γ

Intensity (%)

72.5 - 100.0

50.0 - 72.5

27.5 - 50.0

20.0 - 27.5

12.5 - 20.0

5.0 - 12.5

experimental dataclose to branches withintensities > 30%

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Lattice dynamics of cobalt-deficient CoO

Application of filter to nonstoichiometric CoO

Co0.97O

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

X K Γ L W X Γ 0 0.1 0.20

2

4

6

8

10

12

14

16

18

DOS (1/THz)

Co0.94O

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

X K Γ L W X Γ 0 0.1 0.20

2

4

6

8

10

12

14

16

18

DOS (1/THz)

perturbed phonons with wavelength ∼ size of region disturbed by vacancy

missing Co ⇒ additional O vibrations (new modes)

acoustic branches at small k not affected by vacancies⇓

long wavelength phonons insensitive to point defects

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Lattice dynamics of cobalt-deficient CoO

Oxygens surrounding vacancies

Partial DOS

0.000

0.005

0.010

0.015

0.020

0.025

0.030

2 4 6 8 10 12 14 16 18

pDO

S (

1/T

Hz)

Frequency (THz)

CoO

CoO with 3% vacancies

CoO with 6% vacancies

vacancies affect thehighest frequency LOmodes

no change inlow-frequency acousticregion

Γ-point frequencies

frequency CoO Co0.97O Co0.94O(THz)ωTO 10.25 9.81 9.61ωLO 15.73 15.87 16.08

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Lattice dynamics of cobalt-deficient CoO

Mean-squared displacements vs T

MSD for Co and O

0.000

0.003

0.006

0.009

0.012

0.015

0.018

Mea

n-sq

uare

d di

spla

cem

ents

(10

-10 m

)

CationsCoO

Co0.97O

Co0.94O

γ-ray (Co)

x-ray (Co)

0.000

0.003

0.006

0.009

0.012

0.015

0.018

0 200 400 600 800 1000

Temperature (K)

AnionsCoO

Co0.97O

Co0.94O

γ-ray (O)

x-ray (O)

Co and O at low T5% larger MSD than instoichiometric CoO

negligible differencebetween Co0.97O andCo0.94O

Co and O above RT

Co OCo0.97O 13% 19%Co0.94O 22% 23%

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Lattice dynamics of Fe-doped CoO

Dispersion curves.UFe = UCo = 7.1 eV

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

X K Γ L W X Γ 0 0.1 0.20

2

4

6

8

10

12

14

16

18

DOS (1/THz)

localized modes

mass defect negligible

change in on-site forceconstant at Fe site

at Co site Φ = 10.013 (eV/Å2)const. vs UFe

Force constant at Fe site

UFe (eV) 5.1 6.1 7.1Φ (eV/Å2) 12.264 12.787 13.313

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Dynamics of Fe impurity

Partial phonon DOSUFe = UCo = 7.1 eV

0.000

0.002

0.004

0.006

DO

S (

1/T

Hz)

Fe

Co

0.000

0.010

0.020

0.030

0 2 4 6 8 10 12 14 16 18

Frequency (THz)

Oxygens surrounding Fe

Oxygens surrounding Co

splitting of ωTO

modes corresponding to:oxygens vibrating aroundCo

ωTO = 10.51 THzconst. vs UFe

oxygens vibrating around Fe

oxygens neighboring Fe

UFe (eV) 5.1 6.1 7.1

ωTO (THz) 9.67 9.68 9.72

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Debye-Waller factors in Fe-doped CoO

Vibrational amplitudes

0.000

0.003

0.006

0.009

0.012

0.015

0.018

Mea

n-sq

uare

d di

spla

cem

ents

(10

-20 m

2 ) Cations FeCo in CoO

Co in (Fe)CoO57Fe

γ-ray (Co)x-ray (Co)

0.000

0.003

0.006

0.009

0.012

0.015

0.018

0 200 400 600 800 1000

Temperature (K)

Anions O in CoOO in (Fe)CoO

γ-ray (O)x-ray (O)

MSDFe < MSDCo < MSDO

5% increase in MSDFe withdecreasing UFe

Low T (10 K)

MSDCo > MSDFe by 7%mass + force constant defect

MSDCo , MSDO in Co(Fe)O are4% higher than respective MSDsin CoO(const. vs UFe)

Above RT ( UFe = 7.1 eV)

MSDFe slopecalc. = exp. Co(57Fe)O

Co(Fe)O Co(57Fe)O CoO

ΘD (K) 440 440 500

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Electronic structure of Co(Fe)O and Co(Fe,V)O

Valence and spin states of Fe in CoO and CoO0.97O

Experiment - M ossbauer spectroscopy

2 singlets @ RTferrous line Fe2+

ferric line Fe3+

vicinity of TN = 293 Kmagnetically split component due to Fe2+

T = 78 K magnetically split Fe2+ and Fe3+

G. K. Wertheim, Phys. Rev. 124, 764 (1961)

MethodFP-(L)APW+lo (WIEN2k code)Full-potential(linearized) augmented plane-wave plus local orbitals⇓

Isomer shift and EFG on Fe in Co(Fe)O and Co(Fe,V)O

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Electronic structure of Co(Fe)O and Co(Fe,V)O

Charge and spin states

different Fe-V complexes in CoO considered

negligible changes vs UFe

α = −0.291a.u.3mms−1

Q = +0.17 bU.D. Wdowik and K. Ruebenbauer, PRB 76, 155118 (2007)

Results

Co(Fe)O Co0.97(Fe)O Exp.[6]MCo (µB) 2.74 2.74 Co2+

3.13 Co3+

MFe (µB) 3.67 4.27IS (mm/s) -1.031 -0.371 -1.1245(8) Fe2+

-0.37(5) Fe3+

EFG (mm/s) 0.27 0.28 00.3-0.4 mm/s linewidth

η 0.03 0.07 0

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Electronic structure of Co(Fe)O and Co(Fe,V)O

Site preferences

Fe-V complex in CoO

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

∆E (

meV

/ato

m)

d (10-10 m)

Trivalent and divalent impurities invacancy-defected CoO

-55.0

-45.0

-35.0

-25.0

-15.0

-5.0

5.0

15.0

25.0

35.0

45.0

55.0

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

∆E (

meV

/ato

m)

Cation radius (10-10 m)

Al3+ Ga3+ Fe3+ Mg2+ In3+

First neighbor site

Second neighbor site

magnetic impurities

Fe3+ in 1st n.s.

Co3+ in 2nd n.s.exception: Fe3+ in 2nd n.s.

non-magnetic impurities

’small’ trivalent cations in 2nd n.s.

’large’ trivalent cations in 1st n.s.

divalent cations in 2nd n.s.

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Electronic structure of Co(Fe)O and Co(Fe,V)O

Density od states for Co(Fe)O and Co0.97(Fe)O

DOS of Fe-doped CoO

-60

-40

-20

0

20

40

60

-6 -4 -2 0 2 4 6

DO

S (

stat

es/e

V)

Energy (eV)

CoFeO

DOS of Fe-doped Co0.97O

-120

-80

-40

0

40

80

-6 -4 -2 0 2 4 6

DO

S (

stat

es/e

V)

Energy (eV)

CoFeO

Co3+ and Fe3+ states inside gap

Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Summary and conclusions

1 DFT+U can predict correct ground states of CoO2 Trivalent Co are found in vacancy-defected CoO3 Filter allows to present phonon-dispersion curves to be more close to

the dispersion relations of a real defected sample4 Point defects influence mainly optical phonon region. Long-wavelength

acoustic phonons are practically not affected by defects, i.e.,smallconcentration of defects does not disturb those crystal properties whichare due to the low-frequency acoustic phonons

5 Average MSDs of ions increase with increasing vacancy concentration(decreased intensity of scattered radiation)

6 Differences in the vibrational dynamics of a dopant and host atomsarise from the dirrefence in their force constants

7 Divalent Fe is found in the host CoO lattice undisturbed by cobaltvacancies

8 Trivalent Fe is found in the host CoO lattice decorated with cobaltvacancies

9 Details can be found in PRB 75, 104306 (2007); PRB 77, 115110 (2008); PRB 78, 224114;

J.Phys.:Condens. Matter 21, 125601 (2009)